Power series solution - are these Recurrence relationships correct?

I'm working with solving an ODE via power series, and the recurrence relations seem like they're more complicated than they should be. Can someone check what I've done so far and let me know if I've screwed up royally somewhere?

It's that first recurrence relationship for that seems odd. The RR for the orphan terms has always evaluated to something simple, and certainly never with three levels of terms, nor with making an appearance.